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Theorem 1.1 is known as the Riemann-Lebesgue theorem. However, in terms of 0 0 the (nonhomogeneous) Besov spaces B1∞ and B∞1, we can refine Theorem 1.1 as follows: Theorem 1.2. 1 0 (1) F maps L continuously to B∞1. 0 ∞ (2) F maps B1∞ continuously to L . 0 ∞ 0 1 Since B∞1 is embedded into L continuously and B1∞ is embedded into L continuously, Theorem 1.2 refines Theorem 1.1 in two different directions. We prove Theorem 1.2 in Section 3.5. Let us give another example. Theorem 1.3. Let 1 ≤ p, q ≤ ∞ and s> 0. For j =1, 2,...,n, we define the j-th Riesz transform by: (x − y )f(y) j j Rn Rj f(x) ≡ lim n+1 dy (x ∈ ). ε→0 Rn |x − y| Z \B(x,ε) ˙ s Then Rj maps Bpq continuously to itself. 1 1 ˙ 0 This is a big achievement; Rj does not map L into L continuously. Since B12 1 ˙ 0 1 is almost the same as L , B12 nicely substitutes for L . We prove Theorem 1.3 in Section 5.1. 1.1.2. Special cases. Many important function spaces are realized as special cases of Besov spaces and Triebel-Lizorkin spaces, which are variants of Besov spaces. We ˙ s refer to Sections 4.1 and 4.2 for the definition of the homogeneous Besov space Bpq ˙ s with 0 < p,q ≤ ∞ and s ∈ R and the homogeneous Triebel-Lizorkin space Fpq with 0 t∆ ′ (1.2) kfkHp ≡ sup |e f| < ∞ (f ∈ S ). t>0 p (3) hp ∼ F 0 for 0 t∆ ′ (1.3) kfkhp ≡ sup |e f| < ∞ (f ∈ S ). t∈(0,1) p (4) Denote by BMO the space of bounded mean oscillations whose norm is given by: 1 1 kfk ≡ sup f(x) − f(y) dy dx BMO |B| |B| B:balls ZB ZB HOMOGENEOUSBESOVSPACES 3 1 for f ∈ Lloc. Then we have ˙ 0 (1.4) BMO ∼ F∞2, (5) Denote by bmo the space of local bounded mean oscillations whose norm is given by: kfkbmo ≡ sup |f(x)| dx B:balls,|B|=1 ZB 1 1 + sup f(x) − f(y) dy dx. |B| |B| B:balls,|B|≤1 ZB ZB Then we have 0 (1.5) bmo ∼ F∞2. 1.1.3. Quantity and quality of functions. We can easily grasp the meaning of the parameters p and s in Besov spaces and Triebel-Lizorkin spaces. As a result, we can deal with the quanitity of functions and the quality of functions separately. 1.2. Homogeneous and nonhomogeneous spaces. Next, we move on to the homogeneous spaces and the nonhomogeneous spaces. Roughly speaking, homo- geneous spaces are function spaces described by a set of partial derivatives of the same order; otherwise the space is nonhomogeneous. Example 1.5. Let m ∈ N and 1 ≤ p ≤ ∞. α (1) The homogeneous Sobolev norm kfkW˙ m,p ≡ k∂ fkp is homogeneous. |αX|=m α (2) The nonhomogeneous Sobolev norm kfkW m,p ≡ k∂ fkp is nonhomo- |α|≤m geneous. X Concerning the homogeneous norms, a couple of helpful remarks may be in order. Remark 1.6. Since differentiation annihilates the polynomials or decreases the order of the polynomials, the homogenous norms do not have complete information of the function f. Despite this remark, we have the following good properties. (1) We can not use the nonhomogeneous norm to describe some properties of functions. For example, the dilation f 7→ f(t·) is a typical one. (2) Although the matters depends on the equations we consider, some invariant quantities can be realized as a sum of homogeneous norms. We need to handle each term elaborately. 1.3. Notation. We use the following notation in this note. Notation . (1) The metric ball defined by ℓ2 is usually called a ball. We denote by B(x, r) the ball centered at x of radius r. Given a ball B, we denote by c(B) its center and by r(B) its radius. We write B(r) instead of B(0, r), where 0 ≡ (0, 0,..., 0). (2) Let E be a measurable set. Then, we denote its indicator function by χE. If E has positive measure and E is integrable over f, Then denote by mE(f) the average of f over E. The symbol |E| denotes the volume of E. 4 YOSHIHIRO SAWANO (3) Let A, B ≥ 0. Then, A . B and B & A mean that there exists a constant C > 0 such that A ≤ CB, where C depends only on the parameters of importance. The symbol A ∼ B means that A . B and B . A happen simultaneously. While A ≃ B means that there exists a constant C > 0 such that A = CB. (4) We define (1.6) N ≡{1, 2,...}, Z ≡{0, ±1, ±2,...}, N0 ≡{0, 1,...}. (5) For a ∈ Rn, we write hai≡ 1+ |a|2. (6) Suppose that {f }∞ is a sequence of measurable functions. Then we write j j=1 p 1 p p ∞ q q (1.7) kfjkLp(ℓq ) ≡ |fj (x)| dx , 0 < p,q ≤ ∞ Rn j=1 Z X and 1 q q ∞ p p (1.8) kfjkℓq (Lp) ≡ |fj (x)| dx , 0 < p,q ≤ ∞ Rn j=1 X Z ′ 1 (7) A distribution f ∈ S is said to belong to Lloc if there exist constants C > 0 1 and N ∈ N and F ∈ Lloc such that |hf, ϕi| = F (x)ϕ(x) dx ≤ C sup (1 + |x|)N |∂αϕ(x)| Rn x∈Rn Z |α|≤N X ∞ 1 ′ 1 ′ for all ϕ ∈ Cc . In this case one writes f ∈ Lloc ∩ S , F ∈ Lloc ∩ S , ′ 1 ′ 1 f ∈ S ∩ Lloc or F ∈ S ∩ Lloc. 2. Schwartz spaces 2.1. Definitions–as sets and as topological spaces. 2.1.1. The Schwartz space S and its dual S′. Here we recall the Schwartz space S together with its topology. Definition 2.1. The Schwartz space S is the subspace of C∞ given by ∞ S = f ∈ C∞ : sup hxiN |∂αf(x)| < ∞ . x∈Rn N\=1 |αX|≤N R The topology of S is the weakest one for which the mapping f ∈ S 7→ pN (f) ∈ is continuous for all N ∈ N, where N α 2 pN (f) ≡ sup hxi |∂ f(x)|, hxi≡ 1+ |x| . x∈Rn |αX|≤N p We admit the following fact: Theorem 2.2. [44, p. 249, Th´eor`eme XII] The Fourier transform maps S into S isomorphically. The proof is well known and omitted. Now we move on to the topological dual space S′. HOMOGENEOUSBESOVSPACES 5 Definition 2.3. One defines S′ ≡{f : S 7→ C : f is linear and continuous }. One equips S′ with the weakest topology so that the mapping f ∈ S′ 7→ hf, ϕi ∈ C is continuous for all ϕ ∈ S. We recall the following fundamental characterization of S′ for later considera- tions. Lemma 2.4. For all f ∈ S′ there exists N ∈ N such that |hf, ϕi| ≤ NpN (ϕ) for all ϕ ∈ S. Proof. Since f is continuous at 0, {ϕ ∈ S : |hf, ϕi| < 1} = f −1({z ∈ C : |z| < 1}) is an open set. Thus there exists N ∈ N and δ > 0 such that 1 ϕ ∈ S : p (ϕ) < ⊂{ϕ ∈ S : |hf, ϕi| < 1}, N N implying |hf, ϕi| ≤ NpN (ϕ) for all ϕ ∈ S. ′ ′ 2.1.2. The Schwartz space S∞ and its dual S∞. The space S and its dual S s are the ingredients of the nonhomogeneous Besov space Bpq for 0 < p,q ≤ ∞ and s s ∈ R, while for the homogeneous Besov space Bpq with 0 < p,q ≤ ∞ and s ∈ R ′ we need S∞ and its dual S∞. α Definition 2.5. One defines S∞ ≡ ψ ∈ S : x ψ(x) dx =0 . n N n R α∈\0 Z The main advantage of defining the class S∞ is that when we are given ϕ ∈ S∞ the function given by ψ = F −1[|ξ|α ·Fϕ] is in S∞. In fact, for ϕ ∈ S ϕ ∈ S∞ if and only if Fϕ vanishes up to the arbitrary order at 0. Definition 2.6. ′ (1) Equip S∞ with the weakest topology so that the evaluation f ∈ S′ 7→ hf, ϕi ∈ C is continuous for all ϕ ∈ S∞. (2) Denote by P the set of all polynomials. Embed P into S′ cannonically; for any α the mononomial xα stands for the distribution ϕ ∈ S 7→ xαϕ(x) dx ∈ C. Rn Z Recall that we can endow the quotient X/ ∼ of a topological space (X, OX ) and its equivalence relation ∼ with a natural topology. Definition 2.7. Let (X, OX ) be a topological space. Denote by ∼ the equivalence relation; 6 YOSHIHIRO SAWANO (1) a ∼ a, (2) a ∼ b implies b ∼ a, (3) a ∼ b and b ∼ c imply a ∼ c. Let p be a projection X to X/ ∼. The strongest topology of X/ ∼ for which p is continuous is called the quotient topology of X/ ∼. Remark that S′/P is a quotient space; [f] = [g] for f,g ∈ S′ if and only if f − g ∈P. 2.2. A fundamental theorem. We shall prove the following theorem: Theorem 2.8. ′ ′ (1) For all f ∈ S∞ there exists F ∈ S such that F |S∞ = f. ′ (2) The linear spaces S /P and S∞ are isomorphic. ′ ′ (3) The mapping R : F ∈ S 7→ F |S∞ ∈ S∞ is continuous. ′ ′ (4) The mapping R : F ∈ S 7→ F |S∞ ∈ S∞ is open, that is, R(U) is an open ′ ′ set in S∞ for all open sets U ⊂ S . 2.2.1. Remarks on Theorem 2.8 and its proof. Here we collect some facts for the proof of Theorem 2.8. Remark 2.9. (1) A direct consequence of this theorem is that ′ ′ [f] ∈ S /P 7→ f|S∞ ∈ S∞ is a topological isomorphism. (2) The continuity of R is clear from the definition. (3) Since S′ is NOT metrizable, one can not apply the Baire category theorem. It seems that there is no literature that allows us to apply a version of the Baire category theorem. (4) It seems that one can not find the proofof the openness of R in any literature before [36]. The proof of (1) and (2) is known in the book [54, 5.1.]. Yuan, Sickel and Yang proved (3) and (4) [64, Proposition 8.1]. However, there was a gap. They used the closed graph theorem. However, it seems unclear that one can use the closed graph theorem. The proof in this note is essentially to close the gap of (4) in the proof of [64, Proposition 8.1]. (5) The idea for the proof is to extend many functionals we deal with to VN or WN carefully, where pN N (2.1) VN ≡ S∞ ⊂ C pN N (2.2) WN ≡ S ⊂ C . 2.2.2. ker R. We specify ker R here. The following lemma is fundamental. Theorem 2.10. Let f ∈ S′. Then the following are equivalent. (1) supp(f) ⊂{0}. (2) f is expressed as the following finite sum: λ f = cλ∂ δ0, λX∈Λ n λ where Λ ⊂ N0 is a finite set and ∂ δ0 denotes the distribution defined by λ |λ| λ h∂ δ0, ϕi≡ (−1) ∂ ϕ(0) for ϕ ∈ S. Via the Fourier transform, we can prove: HOMOGENEOUSBESOVSPACES 7 Corollary 2.11. ker(R)= P. Proof of Theorem 2.10. ◦ Suppose supp(f) ⊂{0}. By Lemma 2.4, we can find N ∈ N such that (2.3) |hf, ϕi| ≤ N sup hxiN |∂αϕ(x)| x∈Rn |αX|≤N for all ϕ ∈ S. Choose {cα}α∈N0n,|α|≤2N+1 such that α −|x|2 2N+2 (2.4) ψ(x)= ϕ(x) − cαx e = O(|x| ) |α|≤X2N+1 as x → 0. We claim (2.5) hf, ψi =0. Since each cα depends continuously on ϕ, more precisely, λ cα = kα,λ∂ ϕ(0) λX∈Λα n with Λα a finite set in N0 , we have α −|x|2 hf, ϕi = hf, x e icα. |α|≤X2N+1 ◦ Suppose f is expressed as the following finite sum as in (2). Then we have α |α| α h∂ δ0, ϕi = (−1) ∂ ϕ(0) = 0. Thus supp(f) ⊂{0} once we show (2.5). Let us show (2.5). Recall that VN is defined by (2.1). Let τ : R → R be a function such that χ(2,∞) ≤ τ ≤ χ(1,∞). Then lim pN (ψ − τ(j·)ψ)=0 j→∞ by the Leibniz rule, (2.3) and (2.4). Thus, hf, ψi = lim hf, τ(j·)ψi =0 j→∞ since f is supported away from the origin. This proves (2.5). 2.2.3. Surjectivity of R. We aim here to show that R is surjective. To this end ′ we choose f ∈ S∞ arbitrarily. Then similar to Lemma 2.4, we can find N ∈ N such that (2.6) |hf, ϕi| ≤ NpN (ϕ) for all ϕ ∈ S∞. Thanks to (2.6) f extends uniquely to a countinous linear functional g on the normed space VN , i.e. f = g|S∞. We use the Hahn-Banach theorem to extend g to WN to have a continuous linear functional G defined on WN such that G|VN = g and that |hG, Φi| ≤ NpN (Φ) for all Φ ∈VN . 8 YOSHIHIRO SAWANO 2.2.4. Openness of R. Everything hinges on the following observation: Theorem 2.12. Let ϕ1, ϕ2,...,ϕN ∈ S∞ and let ψ1, ψ2,...,ψK ∈ S. Assume that the system {[ψ1], [ψ2],..., [ψK ]} is linearly independent in S/S∞. Let N ′ U ≡ {f ∈ S∞ : |hf, ϕj i| < 1} j=1 \ ′ ′ be a neighborhood of 0=0S∞ ∈ S∞ and N K ′ ′ V ≡ {f ∈ S : |hf, ϕj i| < 1} ∩ {f ∈ S : |hf, ψki| < 1} j=1 \ k\=1 ′ a neighborhood of 0=0S′ ∈ S . Then R(V )= U. To prove Theorem 2.12, we need the following key lemma: Lemma 2.13. Under the assumptions of Theorem 2.12, there exists N ≫ 1 such that the system {[ψ1], [ψ2],..., [ψK ]} is linearly independent in WN /VN . Proof. Write K 2K−1 K 2 S ≡ (a1,a2,...,aK ) ∈ C : |ak| =1 . ( ) Xk=1 We have only to show that K K akψk : (a1,a2,...,aK ) 6= (0, 0,..., 0) ∈ C ∩VN = ∅ ( ) Xk=1 or equivalently K 2K−1 akψk : (a1,a2,...,aK ) ∈ S ∩VN = ∅ ( ) Xk=1 for some large N ≫ n + 1. 2K−1 n For each (a1,a2,...,aK ) ∈ S , there exists α(a1,a2,...,aK ) ∈ N0 such that K α(a1,a2,...,aK ) x aj ψj (x) dx 6=0. Rn j=1 Z X Since the function K K α(a1,a2,...,aK ) (b1,b2,...,bK) ∈ C 7→ x bj ψj (x) dx ∈ C Rn j=1 Z X K is continuous, there exists an open neighborhood U(a1,a2,...,aK ) ⊂ C of the 2K−1 point (a1,a2,...,aK ) ∈ S such that K α(a1,a2,...,aK ) x bj ψj (x) dx 6=0 Rn j=1 Z X HOMOGENEOUSBESOVSPACES 9 2K−1 for all (b1,b2,...,bK ) ∈ U(a1,a2,...,aK ). Since S is compact, we can find a l l l finite collection (a1,a2,...,aK ) for l =1, 2,...,L such that L 2K−1 l l l S ⊂ U(a1,a2,...,aK ). l[=1 Taking L l l l N ≡ n +2+ |α(a1,a2,...,aK )|, Xl=1 we obtain the desired result. We prove Theorem 2.12. In fact, from Lemma 2.13, for any f ∈ U, we can find G ∈ WN so that G|VN = g and that hG, ψki = 0 for all k = 1, 2,...,K. In fact, we have a strictly increasing sequence of closed linear subspaces {Span(VN ∪ ′ K K {ψk}k=1)}K′=1. Apply successively the Hahn Banach theorem to obtain the desired G. Then observe that G|S ∈ R and R(G|S)= f. Thus F = G|S does the job. Remark 2.14. Under the assumption of Theorem 2.12, set N K ′ ′ V0 ≡ {f ∈ S : |hf, ϕj i| < 1} ∩ {f ∈ S : hf, ψki =0}. j=1 \ k\=1 Then R(V0)= U as the above proof shows. Theorem 2.15. Let ϕ1, ϕ2,...,ϕN ∈ S∞ and let ψ1, ψ2,...,ψK˜ ∈ S. Assume that the system {[ψ1], [ψ2],..., [ψK ]} is a maximal family of linearly independent ele- ˜ ments in {[ψ1], [ψ2],..., [ψK˜ ]} in S/S∞, so that K ≥ K. More precisely, K (2.7) ψk =ϕ ˜k + aklψl Xl=1 for some ϕ˜k ∈ S∞ and akl ∈ C, k = K˜ +1, K˜ +2,...,K and l =1, 2,...,K. Let N K ′ ′ U ≡ {f ∈ S∞ : |hf, ϕj i| < 1} ∩ {f ∈ S∞ : |hf, ϕ˜ki| < 1} j=1 \ k\=1 ′ ′ be a neighborhood of 0=0S∞ ∈ S∞ and N K ′ ′ V ≡ {f ∈ S : |hf, ϕj i| < 1} ∩ {f ∈ S : |hf, ψki| < 1} j=1 \ k\=1 ′ a neighborhood of 0=0S′ ∈ S . Then R(U) ⊃ V . ′ Proof. Let f ∈ V . Then we can find a linear functional F ∈ S such that F |S∞ = f and that (2.8) hF, ψki =0 for k =1, 2,..., K˜ according to Remark 2.14. Observe that hF, ψki = hF, ϕ˜ki∈{z ∈ C : |z| < 1} for k = K˜ +1, K˜ +2,...,K from (2.7) and (2.8). Thus R(U) ⊃ V . 10 YOSHIHIRO SAWANO 3. Homogeneous Besov spaces In this section we consider homogeneous Besov spaces. As we will see, justifying the definition is one of the hard tasks. 3.1. Definition. We define homogeneous Besov spaces and justify their definition. Definition 3.1. Let 1 ≤ p, q ≤ ∞ and s ∈ R. Choose ϕ ∈ S so that χB(4)\B(2) ≤ ϕ ≤ χB(8)\B(1). Define 1 ∞ q ˙ s ′ js −1 q Bpq ≡ f ∈ S /P : kfkB˙ s ≡ (2 kF [ϕj ·Ff]kp) < ∞ . pq j=−∞ X ˙ s ′ The space B is the set of all f ∈ S /P for which the quasi-norm kfk ˙ s is finite. pq Bpq A couple of helpful remarks may be in order. Remark 3.2. (1) Since S′/P is a quotient linear space, the expression f ∈ S′/P is not ap- propriate. Instead, one should have written [f] ∈ S′/P, where [f] denotes the class in S′/P to which f belongs. Nevertheless we write f ∈ S′/P by habit. ′ ′ (2) It does not make sense to consider kfkp for f ∈ S ; S is NOT included in 1 −1 ∞ Lloc. However, the distribution F [ϕj ·Ff] is a C -function as is seen from the identity −1 1 −1 n F [ϕj ·Ff](x)= hf, F ϕj (x − ·)i (x ∈ R ). (2π)n −1 (3) Note that f ∈ P if andp only if F [ϕj ·Ff] = 0 for all j ∈ Z. In fact α when f ∈ P, ϕj ·Ff = 0 for all j ∈ Z, since f ∈ Span({∂ δ0}α∈N0n ). If −1 F [ϕj ·Ff] = 0 for all j ∈ Z, then ϕj ·Ff = 0. Choose a test function ∞ τ ∈ Cc . Since τ is supported away from the origin and τ vanishes outside of a bounded set, one has ∞ τϕ τ = j · ϕ , Φ j j=−∞ X where ∞ 2 (3.1) Φ ≡ ϕj . j=−∞ X ∞ It counts that the function ϕj /Φ makes sense as an element in Cc ; compare the size of their support. Also, the expression (3.1) is essentially a finite sum, so that ∞ τϕ hτ, Ffi = j , ϕ ·Ff =0. Φ j j=−∞ X D E Thus, supp(Ff) ⊂{0}, implying that f ∈P. HOMOGENEOUS BESOV SPACES 11 ˙ s The following observation justifies the definition Bpq as a linear space. In fact, ˙ s we chose ϕ so that the norm of Bpq depends on ϕ. But as the following theorem ˙ s ˙ s shows, Bpq is independent of ϕ as a set, which justifies the definition of Bpq. Theorem 3.3. Let 1 ≤ p, q ≤ ∞ and s ∈ R. Let ϕ, ϕ˜ ∈ S satisfy χB(4)\B(2) ≤ ϕ, ϕ˜ ≤ −j −j χB(8)\B(1). Set ϕj :≡ ϕ(2 ·) and ϕ˜j ≡ ϕ˜(2 ·) for j ∈ Z. Then we have 1 1 ∞ q ∞ q (2jskF −1[ϕ ·Ff]k )q ∼ (2jskF −1[˜ϕ ·Ff]k )q j p j p j=−∞ j=−∞ X X for all f ∈ S′. Proof. By symmetry, it suffices to show that 1 1 ∞ q ∞ q (3.2) (2jskF −1[ϕ ·Ff]k )q . (2jskF −1[˜ϕ ·Ff]k )q . j p j p j=−∞ j=−∞ X X ∞ Define ψ ∈ Cc by ϕ ψ ≡ . ϕ˜−1 +ϕ ˜ +ϕ ˜1 Then we have ϕj = ψj (˜ϕj−1 +ϕ ˜j +ϕ ˜j+1) for each j ∈ Z. Inserting this relation into the right-hand side of (3.2), we obtain 1 ∞ q (2jskF −1[ϕ ·Ff]k )q j p j=−∞ X 1 ∞ q = (2jskF −1[ψ · (˜ϕ +ϕ ˜ +ϕ ˜ ) ·Ff]k )q . j j−1 j j+1 p j=−∞ X By the relation between the Fourier transform, the convolution and the pointwise multiplication, we have 1 ∞ q (2jskF −1[ϕ ·Ff]k )q j p j=−∞ X 1 ∞ q ≃ (2jskF −1ψ ∗F −1[(ϕ ˜ +ϕ ˜ +ϕ ˜ ) ·Ff]k )q . j j−1 j j+1 p j=−∞ X 12 YOSHIHIRO SAWANO By the Young inequality and the fact that kFϕjk1 = kFϕ1k1 for all j ∈ N, we obtain 1 ∞ q (2jskF −1[ϕ ·Ff]k )q j p j=−∞ X 1 ∞ q . (2jskF −1ψ k kF −1[(ϕ ˜ +ϕ ˜ +ϕ ˜ ) ·Ff]k )q j 1 j−1 j j+1 p j=−∞ X 1 ∞ q ≃ (2jskF −1[(ϕ ˜ +ϕ ˜ +ϕ ˜ ) ·Ff]k )q . j−1 j j+1 p j=−∞ X Finally, by the triangle inequality and the index shiftings, we have 1 ∞ q (2jskF −1[ϕ ·Ff]k )q j p j=−∞ X 1 1 ∞ q ∞ q . (2jskF −1[˜ϕ Ff]k )q + (2jskF −1[˜ϕ ·Ff]k )q j−1 p j p j=−∞ j=−∞ X X 1 ∞ q + (2jskF −1[˜ϕ Ff]k )q j+1 p j=−∞ X 1 ∞ q ≃ (2jskF −1[˜ϕ ·Ff]k )q , j p j=−∞ X which proves (3.2). Before we conclude this section, we have a remark helpful till the end of this section. ∞ n Remark 3.4. Let ζ ∈ Cc (R \{0}). The above proof shows that we have many ˙ s possibilities of ζ in the definition of Bpq; 1 ∞ q js −1 q kfkB˙ s ≡ (2 kF [ζj ·Ff]kp) . pq j=−∞ X A particular choise of ζ; ζ = ψ − ψ(2·), where χB(1) ≤ ψ ≤ χB(2), is useful for later considerations. 3.2. H¨older-Zygmund spaces and Besov spaces. So far, we justified the def- ˙ s inition of the homogeneous Besov spaces Bpq with 1 ≤ p, q ≤ ∞ and s ∈ R as a linear space (or a normed space). However, it is rather hard to show that Besov ˙ s space Bpq is complete. As is often the case, it is a burden to construct the limit point when we are given a Cauchy sequence in metric spaces. The main aim of this section is twofold; one is to create a tool to obtain a ˙ s candidate of the limit point when we are given a Cauchy sequence in the space Bpq. HOMOGENEOUS BESOV SPACES 13 The other is to exhibit an example showing that the homogeneous Besov space ˙ s B∞∞ is useful. In fact, as a special case the homogeneous H¨older-Zygmund spaces are realized with the scale B˙ . 3.2.1. The difference operator. To define homogeneous H¨older-Zygmund spaces, we need the notion of difference of higher order. We start with the following ele- mentary identity: m l m m Lemma 3.5. (−1) l mCl = (−1) m! for all m ∈ N. Xl=1 Proof. Compare the coefficient of tm of the function m m t m lt m−l t + ··· = (e − 1) = e mCl(−1) . Xl=0 Then we have m lm 1= (−1)l+m C , m! m l Xl=1 as was to be shown. n n m Let y ∈ R and f : R → R be a mapping. Define inductively ∆y f by 1 m+1 m 1 ∆yf ≡ f(· + y) − f, ∆y f ≡ ∆y (∆mf). Based on Lemma 3.5, we shall obtain a formula to connect the difference with ′ −1 the operator f ∈ S 7→ F [ϕj ·Ff]. Set m m rm · Φ ≡ C · C (−1)r+l+m+1Ψ (rl)n m r m l rl r=1 X Xl=1 1 ∞ Lemma 3.6. Let f ∈ Lloc and Ψ ∈ Cc . Then m jn j r+m+1 m m 2 Φ(2 ·)∗f(x)−m! Ψ(y) dy×f(x)= (−1) r Ψ(y)∆2−j ryf(x) dy. Rn Rn r=1 Z X Z Proof. First we observe that m m m r+l+1+m Φ(y) dy = r (−1) mCr · mCl Ψ(y) dy Rn Rn Z r=1 l=1 Z X X m m m+1 m r l = (−1) r (−1) mCr × (−1) mCl × Ψ(y) dy Rn r=1 X Xl=1 Z = m! Ψ(y) dy Rn Z 14 YOSHIHIRO SAWANO using Lemma 3.5. Thus, it follows that 2jnΦ(2j ·) ∗ f(x) − m! Ψ(y) dy × f(x) Rn m m Z m r+l+m+1 −j = r mCr · mCl(−1) Ψ(y)f(x − 2 rly) dy Rn r=1 X Xl=1 Z − m! Ψ(y)f(x) dy Rn m mZ m r+l+m+1 −j = r mCr · mCl(−1) Ψ(y)f(x − 2 rly) dy Rn r=1 l=0 Z Xm X m r+m+1 m −j = r mCr(−1) Ψ(y)∆2−j ryf(x − 2 rly) dy, Rn r=1 X Z as was to be shown. The equivalent expression in the next lemma is useful when we consider the difference operator. Lemma 3.7. Suppose that Ψ ∈ S satisfies χB(4)\B(2) ≤FΨ ≤ χB(8)\B(1). Define Φ by: m m rm · Φ ≡ C · C (−1)r+l+m+1Ψ . (rl)n m r m l rl r=1 X Xl=1 (1) The function FΦ is constant in a neighborhood of the origin. (2) Let 1 ≤ p, q ≤ ∞ and s ∈ R. Then by setting −j−1 −j ϕj ≡FΦ(2 ·) −FΦ(2 ·) (j ∈ Z) and 1 ∞ q † js −1 q kfk ˙ s ≡ (2 kF [ϕj ·Ff]kp) , Bpq j=−∞ X ˙ s we obtain a norm equivalent to Bpq. Proof. (1) Just observe m m m r+l+m+1 FΦ ≡ r mCr · mCl(−1) FΨ(rl·). r=1 X Xl=1 (2) This assertion follows from Remark 3.4. HOMOGENEOUS BESOV SPACES 15 s 3.2.2. The space C . Denote by Pr the set of all polynomials f ∈P having degree at most r. Definition 3.8. The H¨older space C˙s is the set of all continuous functions f on Rn for which the semi-norm [s+1] k∆ fk ∞ y L Rn kfkC˙s ≡ sup s : y ∈ < ∞. ( |y| ) ˙s Sometimes one considers C modulo P[s]. n Examples 3.9. Denote by CR the set of all complex-valued functions defined on Rn and by C the set of all continuous functions defined on Rn. (1) Let 0 (3.3) χB(1) ≤ ψ ≤ χB(2). Define −j −j+1 (3.4) ϕj ≡ ψ(2 ·) − ψ(2 ·) for j ∈ Z. Set ∞ −1 (3.5) H ≡ F [ϕj ·Ff]. j=1 X It is easy to see that H is a continuous function since ∞ ∞ −js kϕ (D)fk ≤ 2 kfk s ∼kfk s . j ∞ B˙ ∞∞ B˙ ∞∞ j=1 j=1 X X The function H is called the high frequency part of f. Meanwhile, the function ∞ −1 G˜ ≡ F [ϕj ·Ff]. j=1 X is called the low frequency part of f. The trouble is that there is no guarantee that the right-hand side defining G˜ is convergent in some suitable topology. To circumbent this problem, we consider its derivative as follows: 16 YOSHIHIRO SAWANO ˙ s Lemma 3.10. Let f ∈ B∞∞ with s> 0. Let ψ ∈ S satisfy (3.3). Define ϕj by (3.4) for j ∈ Z. If α ≥ [s + 1], then 0 α −1 (3.6) Gα ≡ ∂ F [ϕj ·Ff] j=−∞ X is convergent in L∞ and hence this function is smooth. Proof. Note that −1 −1 F [ϕj ·Ff]= F [(ϕj−1 + ϕj + ϕj+1) · ϕj ·Ff] −1 −1 ≃n F [ϕj−1 + ϕj + ϕj+1] ∗F [ϕj ·Ff] Write τ ≡F −1ϕ. Then we have −1 (j−1)n j−1 jn j (j+1)n j+1 −1 F [ϕj ·Ff] ≃ (2 τ(2 ·)+2 τ(2 ·)+2 τ(2 ·)) ∗F [ϕj ·Ff]. By the H¨older inequality, we have α −1 k∂ F [ϕj ·Ff] k∞ (j−1)(|α|+n) α j−1 j(|α|+n) α j (j+1)(|α|+n) α j+1 = k2 ∂ τ(2 ·)+2 ∂ τ(2 ·)+2 ∂ τ(2 ·)k1 −1 × kF [ϕj ·Ff]k∞ j|α| −1 . 2 kF [ϕj ·Ff]k∞. As a result we have 0 0 α −1 j|α| −1 ∂ F [ϕj ·Ff] ∞ . 2 kF [ϕj ·Ff]k∞ j=−∞ j=−∞ X X 0 j|α|−js ≤ 2 kfk s B˙ ∞∞ j=−∞ X 0 j([s+1]−s) = 2 kfk s B˙ ∞∞ j=−∞ X ≃kfk s . B˙ ∞∞ To construct a substitute of the lower part, we need to depend on a geometric n 1 n property of R ; HDR(R ) = 0. Applying this fact, we can prove the following: ∞ n Lemma 3.11. Let N ∈ N. Suppose that we have {fα}α∈N0 ,|α|=N ⊂ C satisfying ∂βfα = ∂β′ fα′ ′ ′ n ′ ′ ′ for all α, α ,β,β ∈ N0 with |α| = |α | = N and α + β = α + β . Then there exists ∞ α f ∈ C such that ∂ f = fα. The proof is in Appendix; see Section 5.2. By using Lemma 3.11, we have the following control of the low frequency part: ˙ s Corollary 3.12. Let f ∈ B∞∞ with s > 0. Let ψ ∈ S satisfy (3.3). Define ϕj by (3.4) for j ∈ Z. There exists a function G ∈ C∞ such that 0 α α −1 (3.7) ∂ G = ∂ F [ϕj ·Ff] j=−∞ X HOMOGENEOUS BESOV SPACES 17 for all α ∈ N0 with |α|≥ [s + 1]. We further investigate the property of G in Corollary 3.12. ˙ s Proposition 3.13. Let s > 0 and f ∈ B∞∞. Choose ψ ∈ S so that χB(1) ≤ ψ ≤ −j −j+1 χB(2). Define ϕj ≡ ψ(2 ·) − ψ(2 ·) for j ∈ Z. (1) Defien G and H by (3.7) and (3.5), respectively. Then f − (G + H) ∈P. (2) G + H ∈ C˙s. (3) If G′ is such that 0 α ′ α −1 ∂ G = ∂ F [ϕj ·Ff] j=−∞ X ′ for all α ∈ N0 with |α|≥ [s + 1], then G − G is a polynomial of order less than or equal to [s]. Proof. α −1 (1) Observe that ∂ F [ϕj ·F(f − G − H)] = 0 for all α ∈ N0 with |α| ≥ [s + 1] and for all j ∈ Z. Thus F −1[ϕ ·F(f − G − H)] = 0 for all j ∈ Z. j Thus f − G − H ∈P. (2) Write F = G + H. We need to show that [s+1] s (3.8) |∆ F (x)| . |y| kfk s . y B˙ ∞∞ By the triangle inequality we have [s+1] −1 −1 (3.9) |∆y F [ϕj ·Ff](x)| . kF [ϕj ·Ff]k∞. Meanwhile, by the mean value theorem we have [s+1] −1 j[s+1] [s+1] −1 (3.10) |∆y F [ϕj ·Ff](x)| . 2 |y| kF [ϕj ·Ff]k∞. Let j0 ∈ Z be chosen so that (3.11) 1 ≤ 2j0 |y| < 2. We define ∞ −1 Hj0 ≡ F [ϕj ·Ff], Gj0 ≡ G + H − Hj0 . j=j X0 Once we prove [s+1] s (3.12) |∆ G (x)| . |y| kfk s y j0 B˙ ∞∞ and [s+1] s (3.13) |∆ H (x)| . |y| kfk s , y j0 B˙ ∞∞ then we have (3.8). To prove (3.12), we use the mean-value theorem and (3.10) to have [s+1] [s+1] [s+1] |∆y Gj0 (x)| . |y| k∇ Gj0 k∞ j0−1 = |y|[s+1] ∇[s+1] F −1[ϕ ·Ff] j j=−∞ X ∞ j0−1 [s+1] [s+1] −1 ≤ |y | ∇ F [ϕj ·Ff] . ∞ j=−∞ X 18 YOSHIHIRO SAWANO Arguing as we did in Lemma 3.10, we have j0−1 [s+1] j[s+1] [s+1] −1 |∆y Gj0 (x)| . 2 |y| kF [ϕj ·Ff]k∞ j=−∞ X j0−1 j([s+1]−s) [s+1] ≤ 2 |y| kfk s B˙ ∞∞ j=−∞ X j0([s+1]−s) [s+1] ∼ 2 |y| kfk s B˙ ∞∞ s ∼ |y| kfk s B˙ ∞∞ using (3.11). To prove (3.13) we use (3.9) to have ∞ ∞ [s+1] [s+1] −1 −js s |∆ G (x)|≤ 2 kF [ϕ ·Ff]k . 2 kfk s ∼ |y| kfk s , y j0 j ∞ B˙ ∞∞ B˙ ∞∞ j=j j=j X0 X0 as was to be shown. −1 ′ ′ (3) As we did in (1), we F [ϕj ·F(G −G)] = 0. Thus, G−G is a polynomial. α ′ α ′ Since ∂ G = ∂ G for all α ∈ N0 with |α|≥ [s + 1], G − G is a polynomial of order less than or equal to [s]. s ˙ s 3.2.4. The isomorphism C ∼ B∞∞. We can summarize the above observations as follows: Theorem 3.14. Let s > 0. Choose an auxiliary ψ ∈ S so that χB(1) ≤ ψ ≤ χB(2). Set ϕ ≡ ψ(2−1·) − ψ. ˙s ′ ˙s .[C ֒→ S∞, where C is the quotient linear space modulo P[s (1) (2) For F ∈ C˙s, define f : ρ ∈ S∞ 7→ F (x)ρ(x) dx ∈ C. Rn Z ˙ s Then f ∈ B∞∞. ˙ s ˙s (3) Let f ∈ B∞∞. Then there exists a continuous function F ∈ C such that f −F ∈P. More precisely, F can be taken as a sum of continuous functions G and H so that 0 ∞ α α ∂ G = ∂ [ϕj (D)f], H = ϕj (D)f. j=−∞ j=1 X X Proof. (1) Suppose that f ∈ C˙s, where we do NOT consider C˙s as the quotient space. Write m ≡ [s + 1]. For x ∈ R, let us set m m−l (−1) mCl · f(ky + ly) (k ≥ 0), l=0 Xm gk(y) ≡ m−l (−1) mCl · f(ky + ly) (−m ≤ k< 0), l=X−k 0 (k< −m) n for y ∈ R . HOMOGENEOUS BESOV SPACES 19 Then we have k lm l2 f(ky)= ··· gk(y). lmX=−m lm−X1=−m l1X=−m For each x with |x| > 1, choose k ∈ N so that k< |x|≤ k + 1. Set x = ky. Then we have |y|≥ k−1|x| > 1 and |f(x)| . (1 + k)m ≤ (1 + |x|)m as was to be shown. (2) According to Lemma 3.7 and the analogue of Lemma 3.6, we have m −1 r+m+1 m m m F ϕj ∗ f(x)= (−1) r Ψ(y)(∆2−j−1 ryF (x) − ∆2−j ryF (x)) dy. Rn r=1 X Z Thus m −1 m m |F ϕj ∗ f(x)|≤ |Ψ(y)|(|∆2−j−1ryF (x)| + |∆2−j ryF (x)|) dy Rn r=1 Z X m −j s ≤kF kC˙s |Ψ(y)| · |2 y| dy Rn r=1 Z −js X ≃ 2 kF kC˙s , js −1 n and hence 2 |F ϕj ∗ f(x)| . kF kC˙s for x ∈ R , as was to be shown. (3) This is included in Proposition 3.13. 3.3. Fundamental properties. Here we investigate fundamental properties of homogeneous Besov spaces. Let ψ ∈ S satisfy (3.3). Define ϕj by (3.4) for j ∈ Z. We set 1 ∞ q js −1 q kfkB˙ s ≡ (2 kF [ϕj ·Ff]kp) pq j=−∞ X for f ∈ S′/P as before. ′ If f ∈ S∞, then the notation [f] ∈ S /P makes sense. Here we can say more ˙ s about this and we can present some fundamental examples of elements in Bpq. ˙ s .Theorem 3.15. Let 1 ≤ p, q ≤ ∞ and s ∈ R. Then S∞ ֒→ Bpq We use the following lemma: Lemma 3.16. Let x ∈ Rn and j, k ∈ Z. Then we have 2(j+k)n dy 2min(j,k)n n+1 n+1 ∼ n+1 . Rn j 2 2 k 2 2 min(j,k) 2 2 Z (4 |y| + 1) (4 |x − y| + 1) (4 |x| + 1) 20 YOSHIHIRO SAWANO Proof. The proof is simple; by the Fourier transform and the Plancherel theorem, we have 2(j+k)n dy n+1 n+1 Rn j 2 2 k 2 2 Z (4 |y| + 1) (4 |x − y| + 1) ≃ exp(−ξ · xi − (2−j +2−k)|ξ|) dξ Rn Z (2−j +2−k)n 2min(j,k)n ≃ n+1 ∼ n+1 . ((2−j +2−k)−2|x|2 + 1) 2 (4min(j,k)|x|2 + 1) 2 We now prove Theorem 3.15. Proof of Theorem 3.15. Let θ ∈ S∞. We seek to show 1 ∞ q js −1 q kθkB˙ s = (2 kF [ϕj ·Fθ]kp) < ∞. pq j=−∞ X n −1 −2L Let x ∈ R be fixed. It matters that θ˜ ≡ F (|ξ| Fθ) belongs to S∞ for all L ∈ N and that ∆θ˜ = (−1)Lθ. Then we have −1 jn −1 j F ϕj ∗ θ(x)=2 F ϕ(2 y)θ(x − y) dy Rn Z = (−1)L2jn F −1ϕ(2j y)∆Lθ(x − y) dy Rn Z = (−1)L2jn+2jL (∆LF −1ϕ)(2j y)θ(x − y) dy. Rn Z From Lemma 3.16, we obtain 22jL |F −1ϕ ∗ θ(x)|≤ . j (|x| + 1)n+1 Likewise by lettingϕ ˜(ξ) ≡ |ξ|−2Lϕ(ξ), we obtain 2−2jL 2−(n+1)j−2jL |F −1ϕ ∗ θ(x)| . ≤ . j (2j |x| + 1)n+1 (|x| + 1)n+1 Let L ≫ 1. As a result, 1 ∞ q js+|j|(n+1)−2|j|L q kθkB˙ s . (2 ) < ∞, pq j=−∞ X as was to be shown. ′ Next we aim to consider the role of the parameter s. Let f ∈ S∞. Then ∞ −1 ′ ∞ −1 f = j=−∞ F [ϕj ·Ff] in S∞ since Φ = j=−∞ F [ϕj ·FΦ] in S∞ for all Φ ∈ S∞. Observe that P P ∞ α −1 α (−∆) 2 Φ ≡ F [|ξ| · ϕj ·FΦ], j=−∞ X HOMOGENEOUS BESOV SPACES 21 α where the convergence takes place in S∞. Therefore, we can define (−∆) 2 f by ∞ α −1 α (−∆) 2 f ≡ F [|ξ| · ϕj ·Ff], j=−∞ X ′ where the convergence takes place in S∞. It is not so hard to show that the α definition of (−∆) 2 f does not depend on ϕ satisfying (4.3). α 2 ˙ s ˙ s−α Theorem 3.17. Let 1 ≤ p, q ≤ ∞ and s, α ∈ R. Then (−∆) : Bpq → Bpq is an isomorphism. α 2 ˙ s ˙ s−α Proof. We have only to show that (−∆) : Bpq → Bpq is continuous. Indeed, α − 2 ˙ s−α ˙ s once this is proved, then we have (−∆) : Bpq → Bpq is continuous, which is α the inverse of (−∆) 2 . ˙ s Let f ∈ Bpq. Then we have α k(−∆) 2 fk s−α B˙ pq 1 ∞ q = (2j(s−α)kF −1[|ξ|α · ϕ ·Ff]k )q j p j=−∞ X 1 ∞ q = (2j(s−α)kF −1[(ϕ + ϕ + ϕ ) · |ξ|α · ϕ ·Ff]k )q j−1 j j+1 j p j=−∞ X 1 ∞ q ≃ (2j(s−α)kF −1[(ϕ + ϕ + ϕ ) · |ξ|α] ∗F −1[ϕ ·Ff]k )q . j−1 j j+1 j p j=−∞ X If we use the Young inequality, then we have α k(−∆) 2 fk s−α B˙ pq 1 ∞ q ≤ (2j(s−α)kF −1[(ϕ + ϕ + ϕ ) · |ξ|α]k kF −1[ϕ ·Ff]k )q j−1 j j+1 1 j p j=−∞ X 1 ∞ q ≃ (2jskF −1[ϕ ·Ff]k )q j p j=−∞ X = kfk ˙ s , Bpq as was to be shown. Theorem 3.18. Let 1 ≤ p,q,r ≤ ∞ and s ∈ R. ˙ s ˙ s .If r ≥ q, then Bpq ֒→ Bpr (1) (2) In the sense of continuous embedding ˙ s ˙ s−n/p . Bpq ֒→ B∞q (3.14) Proof. .(This is clear because ℓq(Z) ֒→ ℓr(Z (1) 22 YOSHIHIRO SAWANO ˙ s (2) Let f ∈ Bpq. Then we have kfk s−n/p B˙ ∞q 1 ∞ q = (2j(s−n/p)kF −1[ϕ ·Ff]k )q j ∞ j=−∞ X 1 ∞ q = (2j(s−n/p)kF −1[(ϕ + ϕ + ϕ ) · ϕ ·Ff]k )q j−1 j j+1 j ∞ j=−∞ X 1 ∞ q ≃ (2j(s−n/p)kF −1[(ϕ + ϕ + ϕ )] ∗F −1[ϕ ·Ff]k )q . j−1 j j+1 j ∞ j=−∞ X By the H¨older inequality, we have kfk s−n/p B˙ ∞q 1 ∞ q j(s−n/p) −1 −1 q . (2 kF [(ϕ + ϕ + ϕ )]k ′ kF [ϕ ·Ff]k ) j−1 j j+1 p j p j=−∞ X 1 ∞ q ≃ (2jskF −1[ϕ ·Ff]k )q j p j=−∞ X = kfk ˙ s . Bpq Thus we have the desired result. Theorem 3.19. Let 1 ≤ p, q ≤ ∞ and s ∈ R. ˙ s ′ .∞Bpq ֒→ S (1) ˙ s (2) Bpq is complete. Proof. (1) According to Theorems 3.14 and 3.18, ˙ s ˙ s−n/p ˙ s−n/p ˙s−n/p ′ ∞Bpq ֒→ B∞q ֒→ B∞∞ ∼ C ֒→ S (3.15) ˙ s ′ as long as s > n/p. Thus, when s > n/p, Bpq ֒→ S∞. According to ˙ s ′ .Theorem 3.17, we still have Bpq ֒→ S∞ when s ≤ n/p ∞ (2) According to Theorem 3.17, we may assume s > n/p. Let {fj}j=1 be ˙ s ˙ s →֒ a Cauchy sequence in Bpq. Then, according to (3.15) we have Bpq ˙s−n/p ∞ ˙s−n/p ˙s−n/p C . Thus {fj }j=1 is a Cauchy sequence in C . Since C is ∞ ˙s−n/p ˙s−n/p ′ complete, {fj}j=1 is convergent to f ∈ C . Since C ֒→ S∞ and −1 −1 F [ϕj ·Ffl](x) ≃ hF ϕj (x − ·),fli, we have −1 −1 lim F [ϕj ·Ffl](x)= F [ϕj ·Ff](x). l→∞ HOMOGENEOUS BESOV SPACES 23 By the Fatou theorem we have 1 ∞ q js −1 q kf − fkkB˙ s = (2 kF [ϕj ·F[f − fk]]kp) pq j=−∞ X 1 ∞ q js −1 q ≤ lim inf(2 kF [ϕj ·F[fl − fk]]kp) l→∞ j=−∞ X 1 ∞ q js −1 q ≤ lim inf (2 kF [ϕj ·F[fl − fk]]kp) . l→∞ j=−∞ X ˙ s Thus, by letting k → ∞, we learn fk → f in Bpq. 3.4. Realization. When we consider the partial differential equation, it is not confortable to work on the quotient space. One of the reasons is that the quotient space does not give us any information of the value of functions. Therefore, at least we want to go back to the subspace of S′. Although the evaluation mapping does not make sense in S′, we feel that the situation becomes better in S′ than in ′ ′ S∞ ≃ S /P. Such a situation is available when s is small enough. Theorem 3.20. Let 1 ≤ p, q ≤ ∞. Assume n s< p or n s = , q =1. p −j −j+1 Choose ψ ∈ S so that χB(1) ≤ ψ ≤ χB(2). Set ϕj ≡ ψ(2 ·) − ψ(2 ·). Then 0 ∞ ˙ s −1 ∞ −1 for all f ∈ Bpq, F [ϕj ·Ff] is convergent in L and F [ϕj ·Ff] is j=−∞ j=1 X X convergent in S′. 0 −1 jn/p −1 −1 Proof. We use kF [ϕj ·Ff]k∞ . 2 kF [ϕj ·Ff]kp to show that F [ϕj · j=−∞ ∞ X ∞ −1 Ff] is convergent in L as before. The fact that F [ϕj ·Ff] is convergent j=1 X in S′ is a general fact. Or to show this, we can use the lift operator to have ∞ −L −1 ′ (−∆) F [ϕj ·Ff] is convergent in S for any L ∈ N. In fact, we can check j=1 thatX ∞ ∞ −L −1 j(n/p−2L) −1 k(−∆) F [ϕj ·Ff]k∞ . 2 kF [ϕj ·Ff]kp < ∞ j=1 j=1 X X n as long as 2L> . p 24 YOSHIHIRO SAWANO 3.5. Refinement of the Fourier transform. We supplement and prove Theorem 1.3. Theorem 3.21. 1 ˙ 0 (1) The Fourier transform F maps L into B∞1. ˙ 0 ˙ 0 ′ B∞1 ֒→ BC, where B∞1 can be regarded as the subset of S by way of (2) Theorem 3.20. Proof. We assume that ϕ ∈ S satisfies ∞ supp(ϕ) ⊂ B(4) \ B(1), ϕj = χRn\{0}, j=−∞ X −j n where ϕj (ξ) ≡ ϕ(2 ξ) for j ∈ Z and ξ ∈ R . ∞ ∞ −1 (1) kFfk ˙ 0 = kF [ϕj · f(−·)]k∞ ≤ kϕj · f(−·)k1 = kfk1 where B∞1 j=−∞ j=−∞ we used the originalX Hausdorff-Young theoremX to obtain the inequality. ˙ 0 (2) The assumption f ∈ B∞1 is equivalent to ∞ −1 kF [ϕj ·Ff]k∞ < ∞. j=−∞ X Thus, the series ∞ −1 f = F [ϕj ·Ff] j=−∞ X−1 converges uniformly. Since F [ϕj ·Ff] ∈ BC, it follows that f ∈ BC. From the remark below we see that Theorem 3.21 improves the classical Hausdorff- Young theorem. Remark 3.22. Remark that BUC is a closed subspace of L∞, which is of course equipped with the L∞-norm. ˙ 0 ˙ 0 This implies that BUC 6= B∞1. In fact, the above proof shows that BUC ⊃ B∞1. Assume that equality holds. Then the function fj (x) ≡ max(1, min(−1, jx1)) ∈ BUC ∞ ˙ 0 forms a bounded set. This implies that {fj}j=1 ⊂ B∞1 is a bounded set by virtue ′ of the closed graph theorem. Since fj → 2χ{x1>0} − 1 in S ,2χ{x1>0} − 1 would be ˙ 0 a member in B∞1. This is a contradiction to Theorem 3.21(2). 4. More about function spaces ˙ s 4.1. The homogeneous Besov space Bpq for 0 < p,q ≤ ∞ and s ∈ R. We ˙ s extend Bpq for 0 < p,q ≤ ∞ and s ∈ R. Definition 4.1. Let 0 < p,q ≤ ∞ and s ∈ R. Choose ϕ ∈ S so that χB(4)\B(2) ≤ ϕ ≤ χB(8)\B(1). Define 1 ∞ q ˙ s ′ js −1 q Bpq ≡ f ∈ S /P : kfkB˙ s ≡ (2 kF [ϕj ·Ff]kp) < ∞ . pq j=−∞ X HOMOGENEOUS BESOV SPACES 25 ˙ s ′ The space B is the set of all f ∈ S /P for which the quasi-norm kfk ˙ s is finite. pq Bpq Let 0 <η< ∞. We define the powered Hardy-Littlewood maximal operator M (η) by 1 1 η (4.1) M (η)f(x) ≡ sup |f(y)|η dy . |B(x, R)| R>0 ZB(x,R) ! The powered Hardy-Littlewood maximal operator M (η) comes about in the context of the Plancherel-Polya-Nikolskii inequality; |f(x − y)| . (η) sup n/η M (x) y∈Rn (1 + Ry) for all R> 0 and f ∈ S′ such that supp(f) ⊂ B(R). ˙ s The thrust of considering the spaces Bpq with 0 < p,q ≤ ∞ and s ∈ R is the pointwise product f · g, which corresponds to the fact that Lp · Lq = Lr for any 1 1 1 p,q,r > 0 with r = p + q . ˙ s 4.2. The homogeneous Triebel-Lizorkin space Fpq for 0 Definition 4.2. Let 0 < p,q ≤ ∞ and s ∈ R. Choose ϕ ∈ S so that χB(4)\B(2) ≤ ϕ ≤ χB(8)\B(1). Define 1 ∞ q ˙ s ′ js −1 q Fpq ≡ f ∈ S /P : kfkF˙ s ≡ |2 F [ϕj ·Ff]| < ∞ . pq j=−∞ X p ˙s ′ The space F is the set of all f ∈ S /P for which the quasi-norm kfk ˙ s is finite. pq Fpq To handle the convolution, we use the following theorem: ∞ Theorem 4.3. Let 0 1 1 ∞ q ∞ q q q Mfj .p,q,η |fj | . j=1 j=1 X p X p s 4.3. The nonhomogeneous Besov space Bpq for 0 < p,q ≤ ∞ and s ∈ R. Choose ψ ∈ S so that (4.2) χB(2) ≤ ψ ≤ χB(4). Define −j −j+1 (4.3) ϕj ≡ ψ(2 ·) − ψ(2 ·) for j ∈ N. s Using (4.2) and (4.3), we define the nonhomogeneous Besov space Bpq as follows: 26 YOSHIHIRO SAWANO Definition 4.4. Let 0 < p,q ≤ ∞ and s ∈ R. Define s Bpq 1 ∞ q ′ −1 js −1 q ≡ f ∈ S : kfkBs ≡ kF [ψ ·Ff]kp + (2 kF [ϕj ·Ff]kp) < ∞ . pq j=1 X The nonhomogeneous Besov space Bs is the set of all f ∈ S′ for which the quasi- pq s norm kfkBpq is finite. Here we collect some fundamental properties of Besov spaces, which we prove as we did in this note. Remark 4.5. s (1) The definition of Bpq does not depend on ψ satisfying (4.2). s s (2) Bpq is a complete space, that is, Bpq is a quasi-Banach space when 0 < s p, q ≤ ∞ and Bpq is a Banach space when 1 ≤ p, q ≤ ∞. s ′ .S ֒→ Bpq ֒→ S in the sense of continuous embedding (3) 0 2 (4) B22 = L with norm equivalence. s ˙s ∞ ˙s (5) Define C ≡ C ∩ L , where C is NOT a linear space modulo P[s]. Then, s s s C is a Banach space and B∞∞ = C for s> 0 with norm equivalence. (6) Let 0 (5) Let 0 5.1. Proof of Theorem 1.3. Let ψ ∈ S satisfy (3.3). Define ϕj by (3.4) for j ∈ Z. We adopt the following definition of the Besov norm: 1 ∞ q js −1 q kfk ˙ s ≡ (2 kF [ϕj ·Ff]kp) Bpq j=−∞ X for f ∈ S′/P. Set −1 n ξk|ξ| (ϕj−1(ξ)+ ϕj (ξ)+ ϕj+1(ξ)) (ξ ∈ R \{0}), ψj,k(ξ) ≡ (0 (ξ = 0). Then arguing as before, we have 1 ∞ q js −1 −1 q kRkfk ˙ s = (2 kF [ξk|ξ| ϕj ·Ff]kp) Bpq j=−∞ X 1 ∞ q = (2jskF −1[ψ ϕ ·Ff]k )q j,k j p j=−∞ X 1 ∞ q ≃ (2jskF −1ψ ∗F −1[ϕ ·Ff]k )q j,k j p j=−∞ X 1 ∞ q ≤ (2jskF −1ψ k kF −1[ϕ ·Ff]k )q j,k 1 j p j=−∞ X . kfk ˙ s , Bpq as was to be shown. 5.2. Proof of Lemma 3.11. We prove this lemma by induction on N. If N = 1, 1 n then the result is immediate from the Poincar´elemma, or equivalently, HDR(R )= 0. Let N ≥ 2. Define gjγ ≡ fej +γ if j = 1, 2,...,n and |γ| = N − 1. Let ej ≡ (0,..., 0, 1, 0,..., 0), where 1 is in the j-th lot. Then we have ′ ′ δ δ δ δ ′ ′ ∂ gjγ = ∂ fej +γ = ∂ fej +γ = ∂ gj γ ′ ′ n ′ for all j, j =1, 2,...,n, γ,δ,δ ∈ N0 with |γ| = N − 1 and ej + δ = ej′ + δ . Thus ∞ we are in the position of applying the Poincar´elemma to have gj ∈ C satisfying j gjγ = ∂ gγ for all γ with |γ| = N − 1 and j =1, 2,...,n. 28 YOSHIHIRO SAWANO Let α, α′,γ,γ′ ∈ N satisfy |γ| = |γ′| = N −1 and |α| = |α| > 0 and α+γ = α′ +γ′. ′ Suppose α − ej , α − ej′ ≥ 0. Then we have α α−ej j α−ej α−ej ∂ gγ = ∂ ∂ gγ = ∂ gjγ = ∂ fej +γ and ′ ′ ′ ′ ′ ′ ′ ′ α α −ej j α −ej α −ej ′ ′ ′ ′ ′ ′ ∂ gγ = ∂ ∂ gγ = ∂ gj γ = ∂ fej +γ . ′ ′ ′ α α Since (α − ej ) + (ej + γ) = (α − ej′ ) + (ej′ + γ ), we obtain ∂ gγ = ∂ gγ′ . ∞ γ Thus by the induction assumption to have a function f ∈ C such that gγ = ∂ f. Consequently, if |α| = N, then by choosing j so that ej ≤ α we have fα = ∂j gα−ej = α−ej α ∂j ∂ f = ∂ f. 6. Historical notes 6.1. Function spaces. Let us go back to the study of Sobolev spaces initiated in [47, 48, 49]. Zygmund found out that the difference of the second order is useful in s [65]. The space Wp with s ∈ (0, ∞) \ N and 1 ≤ p ≤ ∞ dates back to [2, 45, 21]. s p The space Λp,∞ is defined by Besov in [5, 6]. Let f ∈ L with 1 ≤ p < ∞. When s 0 s When s = 1, the norm kfkΛp,∞ is given by s 1 kfkΛp,∞ ≡kfkWp + sup k∇[f(· + h) − 2f + f(·− h)]kp. h∈Rn\{0} s When 1 6.5. Theorem 2.2. Theorem 2.2 is due to Schwartz [44, p. 105, Th´eor`eme XII]. 6.6. Theorem 2.8. We can find a detailed proof of Theorem 2.8 (1)–(3) in [64, Proposition 8.1]. We refer to [36, Section 6] for the proof of (4). 6.7. Theorem 2.10. The author was not sure when Theorem 2.10 initially ap- peared. To the best knowledge of the author, we can find it in the textbook [4] without proof. 6.8. H¨older-Zygmund spaces. See [65] for H¨older-Zygmund spaces. 6.9. Theorems 2.12 and 2.15. We refer to [36, Section 6] for the proof. 6.10. Theorem 3.3. See the textbooks [4, 41] for the definition of the Besov space s n Bpq with 1 ≤ p, q ≤ ∞ and s ∈ R as in this book. We can find the definition of s the inhomogeneous space Bpq in [4, Definition 6.2.2] and [41, p. 48, Definition 1]. For the definition of homogeneous Besov spaces see [4, 6.3]. 6.11. Theorem 3.14. Theorem 3.14 is essentially due to Taibleson [51, Theorem 4]. See also [54, p. 90, Theorem]. For s > 2, Cs is a function space applied to elliptic differential equations in the paper [1]. See the textbook of Miranda [35] for the detailed background. We refer to [14, 26, 53] for further details. 6.12. Theorems 3.15 and 3.19. We can find Theorems 3.15 and 3.19 in [54, p. 240, Theorem]. 6.13. Theorem 3.17. We can find Theorem 3.17 in [54, 5.2.3]. 6.14. Theorem 3.18. The embedding (3.14) dates back to Triebel [52] when 1 < p,q < ∞. See also [4, Theorem 6.5.1]. For the general case (3.14) is due to Jawerth [29]. See also [54, p. 129, Theorem]. 6.15. Theorem 3.20. We refer to the paper [7] for more about the case for general s. 6.16. Theorem 4.3. We refer to [16] for Theorem 4.3. See also [19, 50]. 6.17. Definition of function spaces. There is a long history in the definition of the function spaces. So the proof is a little simpler. Triebel used the Fourier s multiplier very systematically in [52, Theorem 3.5] to define Bpq with 1 6.18. Textbooks on Besov spaces. We list [9, 15, 14, 18, 25, 27, 28, 43, 46, 55, 56, 57, 58, 59, 60, 61, 62] as textbooks of function spaces. 30 YOSHIHIRO SAWANO References [1] S. Agmon, A. Doulis and L. Nirenberg, Estimates near the boundary for solu- tions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure. Appl. Math. 12 (1959), 623–727. [2] N. Aronszajn, Boundary values of functions with finite Dirichlet integral, Techn. Report 14, Univ. of Kansas, 1955, 77–94. [3] N. Aronszajn and K. T. Smith, Theory of Bessel potentials, I, Ann. Inst. Fourier 11 (1961), 385–476. [4] J. Bergh and J. L¨ofstr¨om, Interpolation spaces. An introduction. 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